Observe the following statements:
statement I: The radical centre of
`S-=x^2+y^2-1=0`
`S' -= x^2+y^2+6x-2y-1=0`
`S''-=x^2+y^2-12x+4y-1=0` does not exist.
statement II : The radical centre of three circles whose centers are collinear does not exist because radical axes of each pair of circles are parallel.

Options

A. I is true, II is true, But II is not the correct explaination for I
B. I and II are true. II is correct explanation of I
C. I is true, II is false
D. I is false, II is true

Answer

To solve the problem, we need to analyze the given statements regarding the radical center of three circles defined by their equations. Let's break down the solution step by step.

### Step 1: Identify the equations of the circles
We have three circles defined by the following equations:
1. \( S: x^2 + y^2 - 1 = 0 \)
2. \( S': x^2 + y^2 + 6x - 2y - 1 = 0 \)
3. \( S'': x^2 + y^2 - 12x + 4y - 1 = 0 \)

### Step 2: Find the radical axis between the first two circles
To find the radical axis of the first two circles, we subtract the equation of \( S' \) from \( S \):
\[ S - S' = (x^2 + y^2 - 1) - (x^2 + y^2 + 6x - 2y - 1) = -6x + 2y = 0 \]
This simplifies to:
\[ 6x - 2y = 0 \quad \text{(Equation 1)} \]

### Step 3: Find the radical axis between the second and third circles
Next, we find the radical axis between \( S' \) and \( S'' \):
\[ S' - S'' = (x^2 + y^2 + 6x - 2y - 1) - (x^2 + y^2 - 12x + 4y - 1) = 6x + 12x - 6y = 18x - 6y = 0 \]
This simplifies to:
\[ 3x - y = 0 \quad \text{(Equation 2)} \]

### Step 4: Analyze the two equations
From Equation 1 and Equation 2, we have:
1. \( 6x - 2y = 0 \)
2. \( 3x - y = 0 \)

### Step 5: Check for parallelism
To determine if the radical axes are parallel, we can rewrite both equations in slope-intercept form:
- From Equation 1: \( y = 3x \)
- From Equation 2: \( y = 3x \)

Both equations have the same slope, indicating that they are indeed parallel.

### Step 6: Conclusion about the radical center
Since the radical axes of the pairs of circles are parallel, the radical center does not exist. This confirms Statement I is true.

### Step 7: Validate Statement II
Statement II claims that the radical center of three circles whose centers are collinear does not exist because the radical axes of each pair of circles are parallel. Since we have established that the radical axes are parallel, this statement is also true.

### Final Conclusion
Both statements are true, and Statement II correctly explains Statement I. Therefore, the correct answer is that both statements are true and Statement II is the correct explanation for Statement I.